Find the maximum and minimum of the
following functions -
(i) y = 5^3 + 2^2 – 3x
Answers
Step-by-step explanation:
The turning points of a graph
The turning points
WE SAY THAT A FUNCTION f(x) has a relative maximum value at x = a,
if f(a) is greater than any value immediately preceding or follwing.
We call it a "relative" maximum because other values of the function may in fact be greater.
We say that a function f(x) has a relative minimum value at x = b,
if f(b) is less than any value immediately preceding or follwing.
Again, other values of the function may in fact be less. With that understanding, then, we will drop the term relative.
The value of the function, the value of y, at either a maximum or a minimum is called an extreme value.
Now, what characterizes the graph at an extreme value?
The slope is 0
The tangent to the curve is horizontal. We see this at the points A and B. The slope of each tangent line -- the derivative when evaluated at a or b -- is 0.
f '(x) = 0.
Moreover, at points immediately to the left of a maximum -- at a point C -- the slope of the tangent is positive: f '(x) > 0. While at points immediately to the right -- at a point D -- the slope is negative: f '(x) < 0.
In other words, at a maximum, f '(x) changes sign from + to − .
At a minimum, f '(x) changes sign from − to + . We can see that at the points E and F.
We can also observe that at a maximum, at A, the graph is concave downward. (Topic 14 of Precalculus.) While at a minimum, at B, it is concave upward.
A value of x at which the function has either a maximum or a minimum is called a critical value. In the figure --
Critical values
-- the critical values are x = a and x = b.
The critical values determine turning points, at which the tangent is parallel to the x-axis. The critical values -- if any -- will be the solutions to f '(x) = 0.